Degrees of Smallness

Mathematics / Calculus

Degrees of Smallness

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. We shall also have to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness.[1]

Small Compared to What?

Before we fix any rules, let us think of familiar cases. There are 60 minutes in the hour, 24 hours in the day, 7 days in the week. Therefore there are 1440 minutes in the day and 10080 minutes in the week.

Obviously 1 minute is a very small quantity of time compared with a whole week. Our forefathers considered it small compared with an hour, and called it "one minute" — meaning a minute fraction, namely one sixtieth of an hour. When they needed still smaller parts, they divided each minute into 60 still smaller parts, which in Queen Elizabeth's day they called "second minutes" (small quantities of the second order of minuteness). We call them "seconds" — but few people know why.

Orders of Smallness

If we fix upon any numerical fraction as constituting what we call "relatively small," we can easily state fractions of higher degrees of smallness.

First, Second, and Third Order
If for the purpose of time, $\frac{1}{60}$ is called a small fraction, then: - $\frac{1}{60}$ of $\frac{1}{60}$ = $\frac{1}{3600}$ is a **second-order** small quantity - $\frac{1}{60}$ of $\frac{1}{60}$ of $\frac{1}{60}$ = $\frac{1}{216000}$ is a **third-order** small quantity Or if $\frac{1}{100}$ (one per cent) is our "small": - $\frac{1}{100} \times \frac{1}{100} = \frac{1}{10{,}000}$ — second order - $\frac{1}{100} \times \frac{1}{100} \times \frac{1}{100} = \frac{1}{1{,}000{,}000}$ — third order

The key insight: the smaller a small quantity is, the more negligible its higher orders become. A second-order small quantity is always negligible compared to the first-order one, provided the first-order quantity is small enough.

The Flea Hierarchy
Thompson quotes Dean Swift: *"So, Nat'ralists observe, a Flea* *Hath smaller Fleas that on him prey.* *And these have smaller Fleas to bite 'em,* *And so proceed ad infinitum."* An ox might worry about a flea — a small creature of the first order. But a flea's flea, being of the second order of smallness, would be negligible. Even a gross of fleas' fleas would not trouble the ox.

Differentials and Their Orders

Now we can apply this to calculus. We write $dx$ for a little bit of $x$. These things — $dx$, $du$, $dy$ — are called "differentials."

If $dx$ is a small bit of $x$, and relatively small of itself, it does not follow that quantities like $x \cdot dx$ or $x^2 \cdot dx$ are negligible — they are first-order small.

But $dx \times dx = (dx)^2$ would be negligible, being a small quantity of the second order.

: A second-order small quantity — negligible compared to $dx$: A first-order small quantity — small but not negligible

This is the single most important rule in differential calculus: we keep first-order small quantities and throw away second-order (and higher) ones.

The Growing Square

Let us think of $x$ as a quantity that can grow by a small amount, becoming $x + dx$, where $dx$ is the small increment added by growth.

The square of this is:

: The original square — this is what we started with: Two rectangles, each $x$ by $dx$ — first-order small, we KEEP this: The tiny corner square — second-order small, we DISCARD this
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The second term $2x \cdot dx$ is not negligible because it is a first-order quantity. The third term $(dx)^2$ is second-order small — a bit of a bit.

Numerical Example
If $dx = \frac{1}{60}$ of $x$: - The second term: $2x \cdot dx = \frac{2}{60} \cdot x^2 = \frac{1}{30}$ of $x^2$ - The third term: $(dx)^2 = \frac{1}{3600}$ of $x^2$ If $dx = \frac{1}{1000}$ of $x$: - The second term: $\frac{2}{1000}$ of $x^2$ — small but meaningful - The third term: $\frac{1}{1{,}000{,}000}$ of $x^2$ — utterly negligible

So when we want to know how much the square of $x$ has grown, we can write:

: The growth of the square — how much $x^2$ increased: The rate at which $x^2$ grows, relative to $dx$: The small amount $x$ grew by

This is our first differentiation. We have found that when $x$ grows by $dx$, $x^2$ grows by $2x \cdot dx$. The $(dx)^2$ has been discarded as negligible.

Why This Works
This isn't hand-waving. As $dx$ gets smaller, the ratio of the discarded $(dx)^2$ to the kept $2x \cdot dx$ is $\frac{dx}{2x}$, which shrinks toward zero. The approximation gets **better** the smaller $dx$ is — and in the limit, it's exact.

Sources

[1]Thompson, S.P. — Calculus Made Easy, 2nd ed. (1914), Ch. IIhttps://www.gutenberg.org/ebooks/33283